Short Notes: Thermal Properties of Matter

Table of Contents
1. 10.1 Temperature Scales
2. 10.2 Thermal Expansion
3. 10.3 Calorimetry
4. 10.4 Phase Change
5. 10.5 Heat Transfer
6. Important Constants and Typical Values
7. Typical Problem Concepts to Remember
8. Summary
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10.1 Temperature Scales

Temperature is a measure of the thermal state of a body that determines the direction of heat flow between bodies: heat flows from a body at higher temperature to one at lower temperature. Temperature scales provide numerical measures of temperature; common scales are Celsius, Kelvin, and Fahrenheit.

Kelvin is the SI unit of temperature; 0 K (absolute zero) corresponds to -273.15 °C. For thermodynamic calculations use Kelvin for absolute temperature in laws that involve powers of temperature (e.g., radiation).

10.2 Thermal Expansion

Most solids, liquids and gases change their dimensions when temperature changes. The change is usually approximately proportional to the original dimension and to the change in temperature for small temperature ranges. The proportionality constants depend on the material.

Definitions and notes:

• \(\alpha\) is the coefficient of linear expansion (per K or per °C).
• \(\beta\) is the coefficient of area expansion; for most solids \(\beta \approx 2\alpha\).
• \(\gamma\) is the coefficient of volume expansion; for most solids \(\gamma \approx 3\alpha\).
• The relations \(\beta = 2\alpha\) and \(\gamma = 3\alpha\) are approximations valid for isotropic, homogeneous solids and for small temperature changes.
• Applications: thermal stress in constrained bodies, gaps in railway tracks, bimetallic strips in thermostats, expansion joints in bridges and pipelines.

10.3 Calorimetry

Calorimetry is the study and measurement of heat transfer associated with temperature changes and phase changes. The basic relation for sensible heat (heat that changes temperature) is:

\(Q = mc\Delta T\)

Definitions:

• \(Q\) - heat transferred (Joules).
• \(m\) - mass of the substance (kg).
• \(c\) - specific heat capacity (J kg⁻¹ K⁻¹).
• \(\Delta T\) - change in temperature (K or °C).

From the above relation:

\(c = \dfrac{Q}{m\Delta T}\)

Latent heat is the heat required for a phase change at constant temperature. The heat involved is:

\(Q = mL\)

\(L\) is the specific latent heat (J kg⁻¹):

• Latent heat of fusion (melting/freezing): energy required to change unit mass between solid and liquid without temperature change.
• Latent heat of vaporisation (boiling/condensation): energy required to change unit mass between liquid and gas without temperature change.

Heat capacity of a body:

\(C = mc\) (J K⁻¹)

Water equivalent is a useful concept in calorimetry; it is the mass of water that has the same heat capacity as the body:

\(W = \dfrac{mc}{c_{\text{water}}}\)

Typically, the specific heat of water is approximately 4186 J kg⁻¹ K⁻¹ (often rounded to 4200 J kg⁻¹ K⁻¹ in some problems).

Principle of calorimetry (conservation of energy): Heat lost by hot bodies = Heat gained by cold bodies (neglecting heat exchange with surroundings).

10.4 Phase Change

• Latent heat of fusion (ice → water): \(L_{\text{f}} = 3.34\times 10^{5}\ \text{J kg}^{-1}\).
• Latent heat of vaporisation (water → vapour): \(L_{\text{v}} = 2.26\times 10^{6}\ \text{J kg}^{-1}\).
• During a phase change at constant pressure, the temperature remains constant while the latent heat is absorbed or released.
• Common phase-change processes: melting/freezing, boiling/condensation, sublimation/deposition.
• Clausius-Clapeyron concept (qualitative): the boiling temperature depends on pressure; lowering pressure lowers the boiling point.

10.5 Heat Transfer

Heat transfer occurs by three distinct mechanisms: conduction, convection and radiation. Each mechanism has characteristic equations and dependencies.

Conduction

Conduction is the transfer of heat through a medium without bulk motion of the medium. In solids it occurs via lattice vibrations and, in metals, also by free electrons.

The steady-state heat current (rate of heat flow) through a uniform slab of area A and thickness L with temperature difference \(T_1 - T_2\) across it is given by Fourier's law (one-dimensional form):

\(\dfrac{Q}{t} = \dfrac{kA\,(T_1 - T_2)}{L}\)

\(k\) is the thermal conductivity (W m⁻¹ K⁻¹). Materials with high \(k\) (metals) are good conductors of heat; materials with low \(k\) (insulators like wood, air, glass-wool) are poor conductors.

Thermal resistance of a slab is defined as:

\(R = \dfrac{L}{kA}\)

For heat flow through series and parallel combinations of layers the resistances combine as:

These relations are widely used in problems involving multi-layered walls, composite rods and thermal insulation design.

Convection

Convection is heat transfer by bulk motion of a fluid (liquid or gas). Natural (free) convection arises from density differences due to temperature gradients; forced convection is produced by external means (fans, pumps).

Convective heat transfer from a surface is often expressed as:

\( \dfrac{Q}{t} = hA(T_{\text{surface}} - T_{\infty})\)

\(h\) is the convective heat-transfer coefficient (W m⁻² K⁻¹); its value depends on the fluid, flow conditions and geometry.

Radiation

Radiation is energy transfer by electromagnetic waves; it does not require a medium. All bodies emit thermal radiation according to their temperature and emissivity.

Stefan-Boltzmann law for a perfect blackbody of area A at absolute temperature T:

\(P = \sigma A T^{4}\)

\(\sigma\) is the Stefan-Boltzmann constant, \(\sigma = 5.67\times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}\).

For net radiative power exchanged between a body at \(T_1\) and surroundings at \(T_2\) (approximate, assuming black surfaces):

\(P_{\text{net}} = \sigma A\,(T_1^{4} - T_2^{4})\)

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Newton's Law of Cooling

Newton's law of cooling gives an approximate description of the rate at which a body exchanges heat with its environment when temperature differences are not very large:

\(\dfrac{dT}{dt} = -k\,(T - T_0)\)

Here \(T\) is the temperature of the body, \(T_0\) is the ambient temperature and \(k\) is a positive constant that depends on body and environment (related to convective/radiative coefficients and body heat capacity). The solution shows exponential approach of T to \(T_0\).

Important Constants and Typical Values

• Specific heat of water: approximately 4186 J kg⁻¹ K⁻¹.
• Latent heat of fusion of ice: \(3.34\times 10^{5}\ \text{J kg}^{-1}\).
• Latent heat of vaporisation of water: \(2.26\times 10^{6}\ \text{J kg}^{-1}\).
• Stefan-Boltzmann constant: \(\sigma = 5.67\times 10^{-8}\ \text{W m}^{-2}\text{K}^{-4}\).

Typical Problem Concepts to Remember

• Use Kelvin for absolute-temperature expressions (radiation, thermodynamic relations).
• For small ΔT use linear relations \(\Delta L = \alpha L_0 \Delta T\) etc.; for larger ranges check whether coefficients vary with temperature.
• Apply conservation of energy in calorimetry: heat lost = heat gained (account for latent heats where phase change occurs).
• When layers are in series, add thermal resistances; when in parallel, add conductances (reciprocal of resistance).
• Distinguish mechanisms: conduction (through materials), convection (by fluid flow), radiation (electromagnetic). Problems may involve one or more mechanisms simultaneously.

Summary

This chapter summarises basic formulae and concepts of thermal properties of matter: temperature scales and conversions; coefficients and formulae for linear, area and volume expansion; calorimetry relations including specific and latent heat; phase-change facts and latent heat values; and the three modes of heat transfer with key formulae for conduction, convection and radiation. These are essential tools for solving numerical and conceptual problems in thermal physics.